Inclusion in the effective multilevel atom rate equations

5.2 Two-photon processes: formal description

5.2.5 Inclusion in the effective multilevel atom rate equations

5.2.5.1 Formal two-photon decay rates

As mentioned earlier, including two-photon decays from states with n > 2 and Raman scattering events poses a double-counting problem. In principle, to avoid this double counting issue, one should discard “1+1” decays (or decays following an absorption event, which is like a Raman scattering event on resonance) altogether. If one were to pursue this idea, one should not consider thepstates at all anymore (as they are formally only intermediate states in two-photon processes), but consider allsanddstates as “interface states” and allow for two-photon recombinations to the ground state.

The two-photonnl↔1stransition rates would then become:

˙

xnl^(2γ)_1s =−x˙1s^(2γ)_nl =x1sR˜^total_1s,nl−xnlR˜_nl,1s^total, (5.11) where the formal transition rates are given by:

R˜^total_1s,nl≡

Z dΛnl

dν gnl

g1s

e^{h(νn1−ν)/Tr}−1⁻¹fνdν (5.12)

and

R˜^total_nl,1s≡

Z dΛnl

dν

e^{h(ν−νn1)/Tr}−1⁻¹dν, (5.13)

where the integrals run fromνn1/2 toνc. In principle Eq. (5.11)–(5.13), can be included in a standard or effective multilevel atom code, provided one solves simultaneously for the radiation field, using the radiative transfer equation Eq. (5.10).

5.2.5.2 Decomposition into “1+1” transitions and non-resonant contributions

Two-photon decays from higher excited states constitute, however, a correction to the recombination history computed in the standard “1+1” picture, and we would like to implement it as such. We start by formally separating the integrals in Eqs. (5.12) and (5.13) in two contributions: the resonant

pieces, forν ≈νn⁰1, and a non-resonant piece, for frequencies far enough from any resonance. We therefore rewrite, formally:

R˜^total_1s,nl = X

n⁰

R˜_1s,nl^(n0p)+ ˜R1s,nl and R˜^total_nl,1s = X

n⁰

R˜_nl,1s^(n0p)+ ˜Rnl,1s, (5.14)

where the resonant contributions ˜R⁽ⁿ_1s,nl^0p)and ˜R⁽ⁿ_nl,1s^0p)are defined in a similar manner as in Eqs. (5.12) and (5.13), but with the integration being carried over a narrow range ∆ν nearνn⁰1, and ˜R1s,nl and R˜nl,1sare the non-resonant pieces required to complete the total rates. So far the separation is just formal and we have not made any approximation.

5.2.5.3 “1+1” Resonant contribution

We now notice that near a resonanceν ≈νn⁰1, the two-photon differential decay ratedΛnl/dνtakes on the following form (ifn > n⁰):

dΛnl

dν _ν

≈ν_n01

≈ 1 4π²

Anl,n⁰pAn⁰p,1s

(ν−νn⁰1)²+ (Γn⁰p/4π)² =Anl,n⁰p

An⁰p,1s

Γn⁰p

φL(ν−νn⁰1; Γn⁰p), (5.15)

where Γn⁰p is the total inverse lifetime of the staten⁰p, and the Lorentzian profile is given by φL(∆ν; Γ)≡ Γ/(4π²)

∆ν²+ (Γ/4π)². (5.16)

Forn < n⁰, the first coefficient in Eq. (5.15) should be gn⁰p/gnl×An⁰p,nl instead ofAnl,n⁰p. When accounting for the thermal motions of atoms, the Lorentzian profile should be replaced by a Voigt profile. We can now approximate the resonant pieces with the following expressions, valid for both n < n⁰ andn > n⁰:

R˜⁽ⁿ_1s,nl^0p)≈3An⁰p,1sf_νn01Rn⁰p,nl

Γn⁰p

(5.17) and

R˜⁽ⁿ_nl,1s^0p)≈Rnl,n⁰p

An⁰p,1s

Γn⁰p

, (5.18)

where f_νn01 is the photon occupation number averaged over the Voigt profile near the resonance ν ≈ νn⁰1. Eqs. (5.17) and (5.18) are exactly what one would obtain in the “1+1” picture after

“factoring out” the pstates (with a procedure similar to what is used to get rid of the “interior”

states in the EMLA method). Having these resonant rates is exactly equivalent to having optically thin one-photon transitions between thenl and n⁰pstates, with rates Rnl,n⁰p(Tr) and Rn⁰p,nl(Tr),

and optically thick Lyman transitions, with net rate:

˙

xn⁰p_1s=−x˙1s_n0p=An⁰p,1s

3x1sf_νn0p−xn⁰p

. (5.19)

To obtain the net decay rates in the Lyman transitions, one then needs to solve for the radiation field in the immediate vicinity of Lyman resonances. If the frequency region for which two-photon transitions are considered as “resonant” is narrow enough, this can be done in the Sobolev approxi- mation. Indeed, all the relevant conditions are met (see also discussion in Ref. [49]; for more details on the Sobolev approximation, see Section 4.2.4):

First, the two-photon absorption and emission profiles can both be approximated by the same resonance profile Eq. (5.15). This relies on the assumption that the blackbody radiation field varies little across the “resonant” region, and requires for its width to satisfy ∆νTr/h.

Secondly, we argued in Section 5.2.3 that one could assume complete frequency redistribution for resonant scattering near the Doppler core of Lyman resonances. The “resonant” region should therefore not exceed a few Doppler widths.

Finally, if we consider regions in frequency narrow enough around the resonances, we can use the steady-state approximation. This requires ∆ν/ν1.

We can see that considering the “resonant” region around each Lyman resonance to be a few Doppler widths wide meets all the requirements.

An additional assumption required here is that excited states are near Boltzmann equilibrium, which is very accurate at redshifts for which two-photon processes are important. In the Sobolev approximation, and in the limit of large Sobolev optical depth, Eq. (5.19) becomes the standard Lyman decay rate Eq. (3.8), wheref_np⁺ is the photon occupation number incoming on the resonance, preprocessed by two-photon processes and diffusion in the blue damping wing of the line.

The Sobolev approximation is probably the least accurate for Lyαdecays where partial redistri- bution due to resonant scattering is important. However, the large optical depth to two-photon ab- sorptions in the Lyman-αblue damping wing, in conjunction with frequent scattering events, drive the radiation field to the equilibrium value fν = xn⁰p/(3x1s)e^{−h(ν−νn01)/Tm} over several Doppler widths (of the order of 40 Doppler widths, see Fig. 4.1 and accompanying discussion in Section 4.2).

As a consequence the net decay rate in the core of the resonance is very small anyway. We checked that in the presence of two-photon transitions and frequency diffusion, even setting ˙xn⁰p

1s = 0 instead of the expression given by Eq. (5.19) leads to relative changes to the recombination history of at most 7×10⁻⁴. Given that frequency diffusion leads to corrections of a few percent at most to the decay rate in Lyαwhen radiative transfer is treated carefully even at the line center [45], we can be confident that using the Sobolev approximation for the resonant contributions of two-photon decays is accurate to better than 10⁻⁴.

5.2.5.4 “Pure two-photon” non-resonant contribution

In the previous section we discussed how two-photon decays within a few Doppler widths of Lyman resonances can in fact be accounted for in the standard “1+1” picture. To evaluate the non-resonant pieces, ˜R1s,nl and ˜Rnl,1s, we need to solve the radiative transfer equation, Eq. (5.10), to obtain the photon occupation number. The subject of Section 5.3 is to describe our numerical method of solution.

Note that choosing the “resonant” regions to be a few Doppler widths has an additional advan- tage. Since a Doppler width is∼10³ times wider than the natural width of Lyman lines, it is not necessary to account for the pole displacements in the computation of the differential two-photon decay rates in the non-resonant region. In addition, the fraction of two-photon decays that are considered non-resonant will be small (of the order of Γnp/(4π²)/∆ν, where ∆ν is the width of the

“resonant” region). For ∆ν of a few Doppler widths, this fraction is ∼10⁻⁴. This means that the

“pure” two-photon decay rates ˜Rnl,1sare much smaller than the total inverse lifetime of thenlstate, Γnl, which is required to simplify the effective MLA model to an effective four-level atom model as we discussed in Section 3.4.4.

As a final note, we want to emphasize why the final result is independent of the exact boundary between “resonant” and “non-resonant” regions, so long as the resonant regions are a few Doppler widths wide. If one were to increase the width of the “resonant” region, then the “pure” two-photon transition rates ˜Rnl,1s and ˜R1s,nl would decrease, mainly because of the change of the integration region in the blue wings of the resonance – in the red wing, the radiation field has reached near equilibrium with the line and the net rate of decays immediately blueward of line center is very small anyway. This decrease would be nearly exactly compensated by the increase of what is considered as

“1+1” decays, as the photon occupation number incoming on the Lyman resonances,f_np⁺, would be decreased due to the smaller optical depth due to “pure” two-photon absorptions in the blue wing.

Hirata (2008) checked the independence of the result from the exact value chosen for the width of the “resonant” region, and found that even changing this width by a factor of 9 lead to relative changes of at most 4×10⁻⁴ in the recombination history.